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Contents of partitions and the combinatorics of permutation factorizations in genus 0

Authors: S. R. Carrell and I. P. Goulden
Journal: Trans. Amer. Math. Soc. 370 (2018), 5051-5080
MSC (2010): Primary 05A15; Secondary 05A05, 05E05, 14E20
Published electronically: February 21, 2018
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Abstract: The central object of study is a formal power series that we call the content series, a symmetric function involving an arbitrary underlying formal power series $ f$ in the contents of the cells in a partition. In previous work we have shown that the content series satisfies the KP equations. The main result of this paper is a new partial differential equation for which the content series is the unique solution, subject to a simple initial condition. This equation is expressed in terms of families of operators that we call $ \mathcal {U}$ and $ \mathcal {D}$ operators, whose action on the Schur symmetric function $ s_{\lambda }$ can be simply expressed in terms of powers of the contents of the cells in $ \lambda $. Among our results, we construct the $ \mathcal {U}$ and $ \mathcal {D}$ operators explicitly as partial differential operators in the underlying power sum symmetric functions. We also give a combinatorial interpretation for the content series in terms of the Jucys-Murphy elements in the group algebra of the symmetric group. This leads to an interpretation for the content series as a generating series for branched covers of the sphere by a Riemann surface of arbitrary genus $ g$. As particular cases, by suitable choice of the underlying series $ f$, the content series specializes to the generating series for three known classes of branched covers: Hurwitz numbers, monotone Hurwitz numbers, and $ m$-hypermap numbers of Bousquet-Mélou and Schaeffer. We apply our pde to give new and uniform proofs of the explicit formulas for these three classes of numbers in genus 0.

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Additional Information

S. R. Carrell
Affiliation: Department of Combinatorics and Optimization, University of Waterloo, Ontario N2L 3G1, Canada

I. P. Goulden
Affiliation: Department of Combinatorics and Optimization, University of Waterloo, Ontario N2L 3G1, Canada

Keywords: Generating functions, transitive permutation factorizations, symmetric functions, Jucys-Murphy elements, contents of partitions
Received by editor(s): August 5, 2016
Received by editor(s) in revised form: November 11, 2016
Published electronically: February 21, 2018
Additional Notes: The work of the second author was supported by an NSERC Discovery Grant.
Article copyright: © Copyright 2018 American Mathematical Society

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